How do you log a core workout?

By what you were doing: calisthenics, yoga, pilates, etc.

Those would fall under calisthenics.

I made a custom private exercise (under cardio) for core exercises, and assigned the same calorie estimate as the database gives for mild stretching (i.e., not much). This was for relatively mild-exertion kinds of things (plank variations, various swiss ball exercises, etc.): I thought "calisthenics" would be an overstatement of calories for what I was doing.

Once you set up a personal exercise, MFP uses the initial calorie estimate to scale your future uses of the exercise based on your minutes entered, and I believe also scales it by changing bodyweight.